Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The radicals of a semigroup


Author: Rebecca Slover
Journal: Trans. Amer. Math. Soc. 204 (1975), 179-195
MSC: Primary 20M10
DOI: https://doi.org/10.1090/S0002-9947-1975-0369584-6
MathSciNet review: 0369584
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper investigates various radicals and radical congruences of a semigroup. A strongly prime ideal is defined. It is shown that the nil radical of a semigroup is the intersection of all strongly prime ideals of the semigroup. Furthermore, a semigroup with zero element is nil if and only if it has no strongly prime ideals. We investigate the question of when the left and right radical congruence relations of various radicals are equal. Some theorems analogous to theorems concerning the radicals of rings are also proved.


References [Enhancements On Off] (What's this?)

  • [1] B. D. Arendt, Semisimple bands, Trans. Amer. Math. Soc. 143 (1969), 133-143. MR 40 #255. MR 0246986 (40:255)
  • [2] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. I, Math. Surveys, No. 7, Amer. Math. Soc., Providence, R. I., 1961. MR 24 #A2627. MR 0132791 (24:A2627)
  • [3] N. J. Divinsky, Rings and radicals, Math. Exposition, No. 14, Univ. of Toronto Press, Toronto, Ont., 1965. MR 33 #5654. MR 0197489 (33:5654)
  • [4] H.-J. Hoehnke, Structure of semigroups, Canad. J. Math. 18 (1966), 449-491. MR 33 #5762. MR 0197597 (33:5762)
  • [5] -, Über das untere und obere Radikal einer Halbgruppe, Math. Z. 89 (1965), 300-311. MR 31 #3526. MR 0179278 (31:3526)
  • [6] Nathan Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R. I., 1956. MR 18, 373. MR 0081264 (18:373d)
  • [7] D. R. LaTorre, Modular congruences and the Brown-McCoy radical for semigroups, Proc. Amer. Math. Soc. 29 (1971), 427-433. MR 43 #6350. MR 0280631 (43:6350)
  • [8] R. H. Oehmke, On maximal congruences and finite semisimple semigroups, Trans. Amer. Math. Soc. 125 (1966), 223-237. MR 34 #2739. MR 0202880 (34:2739)
  • [9] H. Seidel, Über das Radikal einer Halbgruppe, Math. Nachr. 29 (1965), 255-263. MR 32 #1276. MR 0183800 (32:1276)
  • [10] R. E. Slover, Representations of a semigroup, Trans. Amer. Math. Soc. 120 (1965), 417-427. MR 32 #5765. MR 0188326 (32:5765)
  • [11] E. J. Tully, Jr., Representation of a semigroup by transformations acting transitively on a set, Amer. J. Math. 83 (1961), 533-541. MR 25 #135. MR 0136670 (25:135)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20M10

Retrieve articles in all journals with MSC: 20M10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0369584-6
Keywords: Radicals of a semigroup, radical congruence, strongly prime ideal
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society